Two-step inequalities

Since the variable is the value we are looking for, there is still something in the way. a multiplies , so we have to deal with that before we can have our .

Here is where you have to be careful. We have to divide our whole inequality with that a, but an inequality sign depends on whether it is negative or positive.

As we already said, a single division was required to solve this inequality, but this example required us to remember a very important information: when the variable changes signs, the inequality sign changes to its opposite as well!

for

______________

for

Let’s try it on a few examples:

(, so the inequality sign remains the same)

,

On the number line (again, since it is strictly lesser, number two isn’t included in the solution set):

Example 2:

In this case x is multiplied by (-7) so we should divide it by (-7), but it is a negative number, so the inequality will change from to .

Here we have that is greater or equal than . That means that will be included in the solution set.

It would have been similar if the task was set like this:

(we divide by a negative number so the inequality changes from )

Since it is less than or equal to, the number is included in the solution set, which makes the solution on the number line look like this: